A new set of orthogonal functions and its application to the analysis of dynamic systems

doi:10.1016/J.JFRANKLIN.2005.06.005

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A new set of orthogonal functions and its application to the analysis of dynamic systems

Journal Article•DOI•

A new set of orthogonal functions and its application to the analysis of dynamic systems

Anish Deb¹, Anindita Dasgupta, Gautam Sarkar¹•Institutions (1)

01 Jan 2006-Journal of The Franklin Institute-engineering and Applied Mathematics (JOURNAL OF THE FRANKLIN INSTITUTE)-Vol. 343, Iss: 1, pp 1-26

TL;DR: It has been established with illustration that the TF domain technique is more accurate than the BPF domain technique as far as integration is concerned, and it provides with a piecewise linear solution.

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Abstract: The present work proposes a complementary pair of orthogonal triangular function (TF) sets derived from the well-known block pulse function (BPF) set. The operational matrices for integration in TF domain have been computed and their relation with the BPF domain integral operational matrix is shown. It has been established with illustration that the TF domain technique is more accurate than the BPF domain technique as far as integration is concerned, and it provides with a piecewise linear solution. As a further study, the newly proposed sets have been applied to the analysis of dynamic systems to prove the fact that it introduces less mean integral squared error (MISE) than the staircase solution obtained from BPF domain analysis, without any extra computational burden. Finally, a detailed study of the representational error has been made to estimate the upper bound of the MISE for the TF approximation of a function f ( t ) of Lebesgue measure.

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Journal Article•DOI•

A new representation for inverse fuzzy transform and its application

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Alireza Khastan¹•Institutions (1)

Institute for Advanced Studies in Basic Sciences¹

01 Jul 2017

TL;DR: A new representation formula for basic functions of fuzzy transform is introduced and some new approximating properties of the inverse fuzzy transform are described, using block pulse functions, to present properties of sinusoidal basic functions.

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Abstract: In this study, a new representation formula for basic functions of fuzzy transform is introduced and some new approximating properties of the inverse fuzzy transform are described. In particular, using block pulse functions, we present properties of sinusoidal basic functions. As an application, we present a new fuzzy-based method for numerical solution of nonlinear Fredholm integral equations.

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15 citations

Journal Article•DOI•

Optimal Control of Time Delay Systems via Hybrid of Block-Pulse Functions and Orthonormal Taylor Series

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Mahmood Dadkhah, Mohammad Hadi Farahi¹•Institutions (1)

Ferdowsi University of Mashhad¹

01 Mar 2016-International Journal of Applied and Computational Mathematics

TL;DR: In this paper, a new method to find the optimal control of time delay systems with quadratic performance index is discussed based on hybrid functions, which consists of block-pulse functions and orthonormal Taylor series.

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Abstract: A new method to find the optimal control of time delay systems with quadratic performance index is discussed. The method is based on hybrid functions. The properties of the hybrid functions which consists of block-pulse functions and orthonormal Taylor series are presented. The operational matrices of integration, delay, dual and product are used to reduce the solution of optimal control time delay system to the solution of algebraic equations. Numerical examples are included to illustrate the effectiveness and validity of the technique.

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15 citations

Cites methods from "A new set of orthogonal functions a..."

...Special attentions have been given to applications of Walsh functions [7], block-pulse functions [8–10], Laguerre polynomials [11], Legendre polynomials [12–14], Chebyshev polynomials [15,16], Taylor series [17,18] and Fourier series [19,20]....
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Journal Article•DOI•

Approximation, integration and differentiation of time functions using a set of orthogonal hybrid functions (HF) and their application to solution of first order differential equations

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Anish Deb¹, Gautam Sarkar¹, Anindita Ganguly², Amitava Biswas³•Institutions (3)

University of Calcutta¹, Jadavpur University², Academy of Technology³

01 Jan 2012-Applied Mathematics and Computation

TL;DR: A new set of orthogonal hybrid functions (HF) which evolved from synthesis of sample-and-hold functions (SHF) and triangular functions (TF) is proposed which is used to approximate a time function in a piecewise linear manner with the mean integral square error (MISE) much less than block pulse function based approximation which always provides staircase solutions.

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14 citations

Journal Article•DOI•

Optimal control of a class of non-linear time-delay systems via hybrid functions

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Mahmood Dadkhah¹, Mohammad Hadi Farahi², Aghileh Heydari¹•Institutions (2)

Payame Noor University¹, Ferdowsi University of Mashhad²

17 Aug 2015-Ima Journal of Mathematical Control and Information

TL;DR: By expanding various time functions in the systems as their truncated hybrid functions, the solution of optimal control problem is reduced to algebraic equations and by means of operational matrices of integration, delay and product the solution is reduced.

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Abstract: Optimal control problems for a class of time-delay non-linear systems with quadratic performance index are studied. The properties of the hybrid functions which consist of block-pulse functions and orthonormal Taylor series are presented. By expanding various time functions in the systems as their truncated hybrid functions, we attain to algebraic equations and by means of operational matrices of integration, delay and product we reduce the solution of optimal control problem to the solution of algebraic equations. Illustrative examples are included to demonstrate the validity and applicability of the technique.

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14 citations

Cites methods from "A new set of orthogonal functions a..."

...An approach based on discretization techniques with necessary conditions to obtain approximate optimal control and the state for optimal control problems (with non-linear delay systems) was proposed in Gollman et al. (2009), but achieving the necessary conditions in some problems and the implementation of the approach may be faced with difficulties. Koshkouei et al. (2012) proposed a method based on measure theory, functional analysis and linear programming and extended it in order to optimize a definite objective function, and to design an appropriate optimal control for the non-linear time-delay systems....
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...…Walsh functions are used to solve linear delays control problems, the applications of block-pulse functions in control systems are considered in Deb et al. (2006), Jiang & Schaufelberger (1992), Mohammadzadeh & Lakestani (2015), in Kung & Lee (1983) one can see the use of Laguerre polynomials…...
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...An approach based on discretization techniques with necessary conditions to obtain approximate optimal control and the state for optimal control problems (with non-linear delay systems) was proposed in Gollman et al. (2009), but achieving the necessary conditions in some problems and the implementation of the approach may be faced with difficulties....
[...]
...An approach based on discretization techniques with necessary conditions to obtain approximate optimal control and the state for optimal control problems (with non-linear delay systems) was proposed in Gollman et al. (2009), but achieving the necessary conditions in some problems and the implementation of the approach may be faced with difficulties. Koshkouei et al. (2012) proposed a method based on measure theory, functional analysis and linear programming and extended it in order to optimize a definite objective function, and to design an appropriate optimal control for the non-linear time-delay systems. Recently, Marzban & Hoseini (2015) considered numerical treatment of non-linear optimal control problems. Orthogonal functions (OFs) and polynomial series have received considerable attentions in dealing with various problems of dynamic systems (Marcellan & Assche, 2006). For such kind of problem, the approach is that of converting the underlaying differential equation governing the dynamical system to an algebraic form through the use of an operational matrix of integration which can be uniquely determined based on the particular OFs. Special attentions have been given to applications of orthogonal functions in solving control or optimal control problems. For example, in Lo et al. (2009), Palanisamy & Rao (1983), Walsh functions are used to solve linear delays control problems, the applications of block-pulse functions in control systems are considered in Deb et al....
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...An approach based on discretization techniques with necessary conditions to obtain approximate optimal control and the state for optimal control problems (with non-linear delay systems) was proposed in Gollman et al. (2009), but achieving the necessary conditions in some problems and the implementation of the approach may be faced with difficulties. Koshkouei et al. (2012) proposed a method based on measure theory, functional analysis and linear programming and extended it in order to optimize a definite objective function, and to design an appropriate optimal control for the non-linear time-delay systems. Recently, Marzban & Hoseini (2015) considered numerical treatment of non-linear optimal control problems....
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Journal Article•DOI•

New sinusoidal basis functions and a neural network approach to solve nonlinear Volterra–Fredholm integral equations

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Stefania Tomasiello¹, Jorge Eduardo Macías-Díaz², Alireza Khastan, Zahra Alijani³•Institutions (3)

University of Salerno¹, Autonomous University of Aguascalientes², University of Tartu³

16 Jan 2019-Neural Computing and Applications

TL;DR: An alternative and efficient method based on the formalism of artificial neural networks is discussed and the efficiency of the mentioned approach is theoretically justified and illustrated through several qualitative and quantitative examples.

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Abstract: In this paper, we present and investigate the analytical properties of a new set of orthogonal basis functions derived from the block-pulse functions. Also, we present a numerical method based on this new class of functions to solve nonlinear Volterra–Fredholm integral equations. In particular, an alternative and efficient method based on the formalism of artificial neural networks is discussed. The efficiency of the mentioned approach is theoretically justified and illustrated through several qualitative and quantitative examples.

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13 citations

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References

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Modern control engineering

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Katsuhiko Ogata

01 Jan 1970

TL;DR: This comprehensive treatment of the analysis and design of continuous-time control systems provides a gradual development of control theory and shows how to solve all computational problems with MATLAB.

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Abstract: From the Publisher: This comprehensive treatment of the analysis and design of continuous-time control systems provides a gradual development of control theoryand shows how to solve all computational problems with MATLAB. It avoids highly mathematical arguments, and features an abundance of examples and worked problems throughout the book. Chapter topics include the Laplace transform; mathematical modeling of mechanical systems, electrical systems, fluid systems, and thermal systems; transient and steady-state-response analyses, root-locus analysis and control systems design by the root-locus method; frequency-response analysis and control systems design by the frequency-response; two-degrees-of-freedom control; state space analysis of control systems and design of control systems in state space.

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6,634 citations

Journal Article•DOI•

Zur Theorie der orthogonalen Funktionensysteme

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Alfred Haar

01 Sep 1910-Mathematische Annalen

TL;DR: In der Theorie der Reihenentwicklung der reellen Funktionen spielen die sog. orthogonalen Funktionensysteme eine fuhrende Rolle.

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Abstract: In der Theorie der Reihenentwicklung der reellen Funktionen spielen die sog. orthogonalen Funktionensysteme eine fuhrende Rolle. Man versteht darunter ein System von unendlichvielen Funktionen $\phi_1 (s), \phi_2 (s),\ldots$, die in bezug auf die beliebige mesbare Punktmenge $M$ die Orthogonalitatseigenschaft $\int_{(M)}\phi_p(s)\phi_q(s)ds=0$ ($p eq q, p, q=1,2,\ldots$), $\int_{(M)}(\phi_p(s))^2ds=1$ ($p=1,2,\ldots$) besitzen, wobei die Integrale im Lebesgueschen Sinne genommen sind. acces pdf

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1,877 citations

New-York, 1985

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Jocelyne Bédard

01 Apr 2005

475 citations

Journal Article•DOI•

Identification of continuous-time systems

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G.P. Rao, Heinz Unbehauen¹•Institutions (1)

Ruhr University Bochum¹

01 Jan 1991

TL;DR: Continuous-time model-based system identification as mentioned in this paper is a well-established field in the field of control systems and is concerned with the determination of particular models for systems that are intended for a certain purpose such as control.

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Abstract: System identification is a well-established field. It is concerned with the determination of particular models for systems that are intended for a certain purpose such as control. Although dynamical systems encountered in the physical world are native to the continuous-time domain, system identification has been based largely on discrete-time models for a long time in the past, ignoring certain merits of the native continuous-time models. Continuous-time-model-based system identification techniques were initiated in the middle of the last century, but were overshadowed by the overwhelming developments in discrete-time methods for some time. This was due mainly to the 'go completely digital' trend that was spurred by parallel developments in digital computers. The field of identification has now matured and several of the methods are now incorporated in the continuous time system identification (CONTSID) toolbox for use with Matlab. The paper presents a perspective of these techniques in a unified framework.

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373 citations

Journal Article•DOI•

Walsh operational matrices for fractional calculus and their application to distributed systems

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C.F. Cheng¹, Y.T. Tsay¹, Tao Wu²•Institutions (2)

University of Houston¹, Harvard University²

01 Mar 1977-Journal of The Franklin Institute-engineering and Applied Mathematics

TL;DR: In this paper, the Walsh operational matrix for performing integration and solving state equations is generalized to fractional calculus for investigating distributed systems and a new set of orthogonal functions is derived from Walsh functions.

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Abstract: The Walsh operational matrix for performing integration and solving state equations is generalized to fractional calculus for investigating distributed systems. A new set of orthogonal functions is derived from Walsh functions. By using the new functions, the generalized Walsh operational matrices corresponding to √s, √(s2 + 1), e-s and e-√s etc. are established. Several distributed parameter problems are solved by the new approach.

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207 citations